The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map \Phi^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded contact homology chain complex for a contact form supported by (S,h). The chain map \Phi^+ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism \Phi from "The equivalence of Heegaard Floer homology and embedded contact homology I, II" on subcomplexes defining the hat versions. Algebraic considerations then imply that the map \Phi^+ is a quasi-isomorphism.

[1]  M. Hutchings,et al.  Proof of the Arnold chord conjecture in three dimensions, II , 2010, 1111.3324.

[2]  M. Hutchings,et al.  Proof of the Arnold chord conjecture in three dimensions I , 2011 .

[3]  M. Hutchings,et al.  Sutures and contact homology I , 2010, 1004.2942.

[4]  P. Ghiggini,et al.  Embedded contact homology and open book decompositions , 2010, 1008.2734.

[5]  Miguel Abreu,et al.  New Perspectives and Challenges in Symplectic Field Theory , 2009 .

[6]  C. Taubes,et al.  Periodic Floer homology and Seiberg-Witten Floer cohomology , 2009, 0906.0383.

[7]  C. Taubes.  Embedded contact homology and Seiberg–Witten Floer cohomology II , 2008, 0811.3985.

[8]  K. Honda,et al.  On the flux of pseudo-Anosov homeomorphisms , 2008, 0809.1347.

[9]  W. Kazez,et al.  Contact structures, sutured Floer homology and TQFT , 2008, 0807.2431.

[10]  Paul Seidel,et al.  Fukaya Categories and Picard-Lefschetz Theory , 2008 .

[11]  C. Wendl.  Automatic transversality and orbifolds of punctured holomorphic curves in dimension four , 2008, 0802.3842.

[12]  M. Hutchings,et al.  Gluing pseudoholomorphic curves along branched covered cylinders II , 2007, 0705.2074.

[13]  C. Wendl.  Finite energy foliations on overtwisted contact manifolds , 2006, math/0611516.

[14]  C. Taubes.  The Seiberg–Witten equations and the Weinstein conjecture , 2006, math/0702366.

[15]  W. Kazez,et al.  On the contact class in Heegaard Floer homology , 2006, math/0609734.

[16]  H. Hofer.  Holomorphic curves and dynamics in dimension three , 2006 .

[17]  Robert Lipshitz.  A cylindrical reformulation of Heegaard Floer homology , 2005, math/0502404.

[18]  Michael C. Sullivan,et al.  Rounding corners of polygons and the embedded contact homology of T 3 , 2004, math/0410061.

[19]  C. Taubes,et al.  Seiberg-Witten and Gromov invariants for symplectic 4-manifolds , 2005 .

[20]  Michael C. Sullivan,et al.  The periodic Floer homology of a Dehn twist. , 2004, math/0410059.

[21]  C. Wendl.  Finite Energy Foliations and Surgery on Transverse Links , 2005 .

[22]  D. Dragnev.  Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations , 2004 .

[23]  Dusa McDuff,et al.  J-Holomorphic Curves and Symplectic Topology , 2004 .

[24]  E. Zehnder,et al.  Compactness results in Symplectic Field Theory , 2003, math/0308183.

[25]  Emmanuel Giroux.  Géométrie de contact: de la dimension trois vers les dimensions supérieures , 2003 .

[26]  Boris Khesin,et al.  Symplectic and Contact Topology: Interactions and Perspectives , 2003 .

[27]  M. Hutchings.  An index inequality for embedded pseudoholomorphic curves in symplectizations , 2001, math/0112165.

[28]  P. Ozsváth,et al.  Holomorphic disks and three-manifold invariants: Properties and applications , 2001, math/0105202.

[29]  P. Seidel.  A long exact sequence for symplectic Floer cohomology , 2001, math/0105186.

[30]  P. Ozsváth,et al.  Holomorphic disks and topological invariants for closed three-manifolds , 2001, math/0101206.

[31]  S. Donaldson,et al.  Lefschetz pencils and the canonical class for symplectic four-manifolds , 2000, math/0012067.

[32]  Thomas H. Parker,et al.  The symplectic sum formula for Gromov–Witten invariants , 2000, 1510.06943.

[33]  H. Hofer,et al.  Introduction to Symplectic Field Theory , 2000, math/0010059.

[34]  K. Honda.  On the classification of tight contact structures I , 1999, math/9910127.

[35]  Thomas H. Parker,et al.  Relative Gromov-Witten invariants , 1999, math/9907155.

[36]  C. Taubes.  The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S^1 X B^3 , 1999, math/9901142.

[37]  E. Zehnder,et al.  Properties of pseudoholomorphic curves in symplectisations. I : asymptotics , 1996 .

[38]  E. Zehnder,et al.  Properties of Pseudoholomorphic Curves in Symplectisations Iv: Asymptotics with Degeneracies , 1996 .

[39]  E. Zehnder,et al.  Properties of pseudo-holomorphic curves in symplectisations II: Embedding controls and algebraic invariants , 1995 .

[40]  B. White,et al.  The structure of branch points in minimal surfaces and in pseudoholomorphic curves , 1995 .

[41]  H. Hofer,et al.  On genericity for holomorphic curves in four-dimensional almost-complex manifolds , 1997 .

[42]  M. Kontsevich,et al.  Gromov-Witten classes, quantum cohomology, and enumerative geometry , 1994, hep-th/9402147.

[43]  D. Mcduff.  Singularities and positivity of intersections of J-holomorphic curves , 1994 .

[44]  H. Hofer.  Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three , 1993 .

[45]  Emmanuel Giroux.  Convexité en topologie de contact , 1991 .

[46]  D. Mcduff.  The local behaviour of holomorphic curves in almost complex 4-manifolds , 1991 .

[47]  M. Gromov.  Pseudo holomorphic curves in symplectic manifolds , 1985 .